Here is a description of topics for projects in analysis. The projects can be done both individually on in small groups, the level and precise formulation of the problem will depend on the student's background and interests. Are you interested or just want more info feel free to contact me.

Efficient analysis of signals well concentrated on given intervals in time and frequency is an old and important problem. Beautiful approach was suggested by SLepian, Landau and Pollak in early 1960s in a series of articles in Bell System Technical Journal. In 1980s Daubechies suggested to use joint time-frequency concentration using the short time Fourier transform and defined localization operators whose spectra can be analyzed when the symbol and the window function are spherically symmetric. In practice other localization operators are of interest and this project will study such operators and their spectra.

References:

1. D. Slepian and H. O. Pollak. Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - I Bell System Technical Journal 40 (1961)
2. H. J. Landau and H. O. Pollak. Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - II Bell System Technical Journal 40 (1961)
3. I. Daubechies. Time-Frequency Localization Operators: A Geometric Phase Space Approach IEEE Trans. of Information Theory, vol. 34 (1998)
4. E. Cordero and K. Grochenig. Symbolic Calculus and Fredholm Property for Localization Operators Journ. of Fourier Analysis and Applications, vol. 12 (2006)

The Heisenberg uncertainty principle has a mathematics formulation, which roughly says that a function and its Fourier transform can not be highly localized simultaneously (or a signal can not be localized both in time and frequency). Many versions of this statement are known. The project consist of studying recent results that can be viewed as discrete versions of the uncertainty principles and their applications. '

References:

1. D. L. Donoho and P. B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math., vol. 49, no. 3, pp. 906–931, 1989.
2. M. Elad and A. M. Bruckstein, A generalized uncertainty principle and sparse representation in pairs of bases, IEEE Trans. Inf. Theory, vol. 48, no. 9, pp. 2558–2567, Sep. 2002.
3. E.J.Candès, J.Romberg, Quantitative robust uncertainty principles and optimally sparse decompositions. Found. Comput. Math. 6 (2006), no. 2, 227–254.
4. E.J.Candès, J.Romberg, T.Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory 52 (2006), no. 2, 489–509.

Traditionally harmonic functions in a ball are represented by series of harmonic polynomials. This technique is very well studied and many properties of harmonic polynomials or spherical harmonics are known. In modern calculations it turns out that another bases are more appropriate for different problems. one of the popular approaches is construction of a wavelet basis on the sphere. The method is used in geo-mathematics and image processing and is interesting from the theoretical point of view. The project will give an overview of the known bases and their applications for different problems.

References:

1. P.Schroder, W.Sweldens Spherical wavelets: Efficiently Representing Functions on the Sphere, Proc. Intern. conference on computer graphics and interactive techniques, 1995
2. V.Michel applied and Comp. harmonic analysis, 12, 77-99,2002
3. M.Holschneider, I.Iglewska-Nowak Poisson Wavelets on the sphere, Journ. Fourier Analysis and appl., 13, no.4, 405-419,2007.
4. M.Hayn, M.HolschniederDirectional spherical multipole wavelets Journ. of Math. Physics, 2009